Less than 24 hr layover.

Your itinerary is definitely eligible for 24 hour visa free transit in Xiamen and you don't need any kind of visa. I understand that you have called the airline, but it is also very sad truth that their customer services are mostly so ignorant even about the very basic details that they should know, so you should not expect them to know the details that they always gladly outsource to the travelers as is the subject of visas and similar.

You will print out your flight tickets and have those printouts with you. Also, you will declare at check-in in Los Angeles that you will travel to Manila via Xiamen using 24 hour visa free transit. After a short inspection of your tickets, you will be allowed to board the plane.

Upon arrival in Xiamen, you will approach 24 hour visa free transit counter where your will apply for a temporary entry permit to be allowed to exit the airport and to enter the country. You are advised to book some simple accommodation and to have a confirmation of such accommodation with you. After your layover in Xiamen, you will continue your journey to Manila.